Contributions of the muscular torques and motion-dependent torques to generate rapid elbow extension during overhand baseball pitching

Sports Eng (2008) 11:47 5 DOI 101007/s12283-008-0002-3 ORIGINAL ARTICLE Contributions of the muscular torques and motion-dependent torques to generate rapid elbow extension during overhand baseball pitching K Naito Æ T Maruyama Published online: 19 June 2008 Ó International Sports Engineering Association 2008 Abstract Some studies have reported that overarm baseball pitching shows a proximal to distal sequential joint motion including a rapid extension of the elbow It has been suggested that the rapid elbow extension just before ball release is not due to the action of the elbow extensor muscles, but the underlying mechanisms are not so clear The purpose of this study was to determine the contributions of each joint muscular- and motion-dependent torques, including the upper trunk and throwing arm joints to generate the rapid elbow extension during baseball pitching The right handed throwing motions of three baseball pitchers were recorded using five high-speed video cameras and the positional data were calculated using the direct linear transformation method A throwing arm dynamic model of the upper trunk and throwing arm joints was then used, including 10 degrees of freedom, to calculate the throwing arm joint muscular-, throwing arm and upper trunk joint motion-, gravity-, and external forcedependent components that contribute to the maximum elbow extension angular velocity The results showed that the rapid elbow extension was primarily due to the upper trunk counterclockwise rotation and shoulder horizontal adduction angular velocity-dependent torques This study implied that the trunk counterclockwise rotators and shoulder horizontal adductors generate positive torques to maintain the angular velocities of the upper trunk counterclockwise rotation and shoulder horizontal adduction may play a key role in producing the rapid elbow extension K Naito (&) T Maruyama Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku 152-8552, Japan e-mail: nassk-918@w9dionnejp Keywords Baseball Biomechanics Elbow extension Motion-dependent torque Muscular torque Pitching 1 Introduction The overhand throwing pattern is very complex due to the speed and number of joint actions involved, including sequential use of pelvic rotation, spinal rotation, shoulder medial rotation and elbow extension [14] Some studies have reported that overarm throwing shows a proximal to distal (P D) sequence in order of joint motions [, 7, 14, 15] Especially, it was shown that a rapid elbow extension in baseball pitching commonly occurs during the acceleration phase while shoulder horizontal adduction slows down [, 7, 14] Although some investigators addressed the question of how the P D sequence in the throwing arm joint motion involving elbow extension is produced [, 11, 17], little is known about the underlying dynamic mechanisms Feltner and Dapena [8] analysed the motion-dependent interactions between the throwing upper arm and forearm in baseball pitching That study showed that the rapid elbow extension that occurred before ball release was primarily generated by the upper arm motion-dependent torque (due primarily to the rotation of the trunk), and not by the elbow extensor torque However, because the throwing arm was modelled as a two-segment kinetic chain, the motion-dependent interactions between the multi-joints including the wrist, elbow, shoulder and trunk were not analysed Herring and Chapman [11] simulated an overarm throw using a planar three-segment kinetic chain model including the shoulder, elbow and wrist to determine the optimal sequence of onset of joint torques which gave the maximum range and velocity of the ball

48 K Naito, T Maruyama It was shown that the best throw was achieved when joint motions and the onset of joint torques were in P D sequential pattern, and the authors concluded that the joint torques were generated in a P D temporal sequence which was effective for increasing the ball velocity However, from the experimental results of natural throwing motions of skillful pitchers, temporal onset of the major muscular torques of the trunk and throwing arm joints were not strictly in a P D sequence [14] In the case of planar 2D throwing, the joint axes of trunk and throwing arm are always parallel, while in the case of natural 3D throwing, the directions of the joint axes of trunk and throwing arm are not parallel as regards to the segmental configuration Despite the fact that natural throwing motion includes multi-joint rotations of the whole body to produce the large velocity of the throwing hand, shoulder horizontal adduction and internal rotation, which are thought to be important and were examined in the previous studies [10, 18], are not included in 2D model analysis Because of this incongruity in the experimental results and/or description of throwing motion between 2D and 3D modelling, the relevance of planar 2D analysis to natural throwing motion is not fully acceptable The linked system of the thrower s body acts as open chain, and thus, the movement of each segment affects other segments [14] In a multi-joint limb system the torque at one joint occurs not only from muscular torque acting at that joint but also from interaction torques (the torque resulting from the motion at another joint) Because of this, the question of which joint torques generate the joint angular velocity of the throwing arm cannot be determined by kinematic or kinetic measurement alone To understand the cause of the rapid elbow extension, it is important to analyse the interaction between the elbow joint motion and the muscular- or motion-dependent torque of each joint of the throwing arm and trunk in dynamic terms Although previous investigations have examined the role of interaction torques in the overarm throws [, 12 14], to the best of our knowledge, no studies have been done to explain the motion-dependent interactions of multijoint limb systems, including the throwing arm and trunk joints, resulting in the rapid elbow extension in the dynamic sense The purpose of this study was to determine the contributions of the throwing arm muscular torques, joint rotations and trunk motions of the multi-joint limb system that develops maximum elbow extension angular velocity during baseball pitching In particular, we mainly focus on improving the model for analysing the relationship between a given joint motion and causal factors resulting in that joint motion, and to explain the dynamic mechanism using the experimental results and the model The question of which joint primarily generated the rapid elbow extension via the muscular- or motion-dependent torque was answered by the contributions of each factor within the throwing arm dynamic model Because 3D natural throwing was examined, it is hypothesised that the rapid elbow extension during baseball pitching would largely result from the proximal joint motion-dependent interaction (particularly due to the trunk rotation), as mentioned by Feltner [], rather than the muscular torque-dependent interaction such as in the 2D simulated motion Detailed understanding about the underlying mechanism for producing rapid elbow extension during baseball pitching is discussed with the contributors revealed in the present study 2 Methods 21 Experiment Three subjects volunteered to participate in the experiment Two were collegiate baseball pitchers and another was a recreational one Their mean ± SD for age (years), height (m) and body mass (kg) were 250 ± 8, 1743 ± 0057, 72 ±, respectively Subjects were explained the experimental procedure before the trials, and informed consent was obtained from each subject The experimental procedure was approved by the Ethical Committee of Graduate School of Decision Science and Technology of Tokyo Institute of Technology, Department of Human System Science The pitching mound was set to a 15 cm height from the floor in the gym, and the home plate was located at the position 1844 m far from the pitching rubber on the mound The subjects were instructed to perform maximal effort pitches toward a squatting catcher placed behind the home plate The throwing motions of the subjects were recorded using five high-speed video cameras of two types (FASTCAM-PCI500C, Photron Inc, Tokyo and HSV- 500C 3, Nac Inc, Tokyo) All cameras were set to film at 250 Hz with 1/1,000 s exposure time and were synchronized electronically Two cameras were located to the right and left of the pitchers, along the pitching rubber Two more cameras were located to the right and left of the pitchers and behind him One camera was located to the right of the pitchers and in front of him In order to estimate the joint centre locations, reflective markers were attached at the greater trochanters, lateral tips of the acromion processes, lateral humeral epicondyles, the radial and ulnar styloid processes of the throwing arm and the head of the third metacarpal on the dorsal aspect of the throwing hand To locate the wrist joint, a band approximately 2 cm wide was placed around the throwing wrist and reflective

Contributions of the muscular torques during overhand baseball pitching 49 markers were attached at the radial and ulnar styloid processes on the band The recorded data were transformed to positional data with digitising software (Frame DIAS II, DKH Inc, Tokyo), utilizing the direct linear transformation (DLT) method [1] The control object for the DLT method was recorded prior to filming the pitches The 81 control points on the control object were distributed throughout the filming area which encompassed a volume of about 20 m 9 24 m 9 20 m For the global reference frame (R 0 ), the global Y direction was defined as a vector from the centre of the pitching rubber to the centre of the home plate The global Z direction was defined as a vector pointing vertically from the centre of the pitching rubber The global X direction was defined as the cross-product of the Y and Z Positional data were digitally filtered independently in the X, Y and Z directions with a 134 Hz Butterworth lowpass filter [10] Digitised locations for the left hip, right hip, left shoulder, right shoulder, and right elbow markers were translated to estimated joint centre locations [3] Digitised locations of wrist and hand marker locations were also mathematically translated to estimate joint centres for wrist and head of the third metacarpal [3] In each time frame, local reference frames at the upper trunk, the throwing shoulder, elbow, wrist and hand were calculated The segment reference frames were located at the mass centre of each segment The upper trunk, upper arm and forearm reference frames (R 1, R 2 and R 3, respectively) were determined as presented by Hong et al [14], and the hand reference frame (R 4 ) was determined as presented by Barrentine et al [3] Analysis of the trials began at the instant approximately 00 s before stride foot contact, and ended at an instant approximately 00 s after ball release Linear ball velocities of all trials were measured by a radar gun (Speed max II, ZETT Inc, Tokyo) To measure correctly ball speed in the experiment, a radar gun was located behind the catcher (about 20 m distant from the pitching rubber) and along path of the ball From the previous studies, it is known that ball speeds of pitches thrown by high school and collegiate baseball pitchers ranged from 33 to 35 m/s [3, 10] In the present study, to consider the contribution to standard baseball pitching movements, trials in which the ball speed exceeded 120 km/h (about 333 m/s) and crossed the strike zone at home plate were analysed Two pitches from each of the three pitchers (total six) satisfied the two conditions 22 The throwing arm dynamic model The throwing system was modelled as a linked system consisting four rigid bodies (upper trunk, throwing upper arm, forearm and hand) and the ball The model involved the upper trunk rotations about three axes and the throwing shoulder, elbow and wrist joints including 7 degrees of freedom (Fig 1) Each joint angle of the model was calculated using the Eulerian angle convention described by Chao [4] The Eulerian convention is defined as the rotational sequence of motion that follows the order of coordinate axes Z, y 0 and x 00, and the Eulerian angles q 1, q 2 and q 3 represent as rotational angles about Z, y 0 and x 00 of the right-handed reference frame, shown in Fig 2 For the upper trunk, the counterclockwise (+) and clockwise (-) rotation h Y, medial (+) and lateral (-) lean h P, and posterior (+) and anterior (-) lean h R were defined as the Eulerian angles q 1, q 2 and q 3 about Z, y 0 and x 00, respectively The neutral position of the upper trunk reference frame was coincident with the global reference frame in three directions In this case, the unit vectors corresponding to the directions of Z, y 0 and x 00 were defined as the joint axes of the upper trunk counterclockwise/clockwise rotation (s Y ), medial/lateral lean (s P ) and posterior/anterior lean (s R ), respectively For the shoulder, the horizontal adduction (+) and abduction (-) h 1, adduction (+) and abduction (-) h 2, and external (+) and internal (-) rotation h 3 were defined as the Eulerian angles q 1, q 2 and q 3 about Z, y 0 and x 00, respectively The neutral position of the upper arm reference frame was coincident Hand R (x,y,z ) 4 4 4 4 s 5 Forearm R (x,y,z ) 3 3 3 3 R 0 Y s 7 Z O Ball s s 4 l u l h l f l t s R s 3 s 1 s 2 s Y s P Upper trunk R (x,y,z ) 2 2 2 2 R (x,y,z ) 1 1 1 1 Upper arm X Fig 1 Throwing arm dynamic model s R ; s P ; s Y : the joint axes of the upper trunk posterior/anterior lean, medial/lateral lean, counterclockwise/clockwise rotation s 1, s 2, s 3 : the joint axes of the shoulder horizontal adduction/abduction, adduction/abduction, external/internal rotation s 4 : the joint axis of the elbow extension/flexion s 5, s, s 7 : the joint axes of the wrist ulnar/radial flexion, flexion/extension, supination/pronation l t : the vectors pointing from the mid-point of the shoulder line to the throwing shoulder l u, l f, l h : the vectors pointing from the proximal end to the distal end of the upper arm, forearm and hand R 0 : global reference frame R 1, R 2, R 3, R 4 : segment reference frames at the upper trunk, upper arm, forearm and hand The X, Y and Z components in R 1, R 2, R 3 and R 4 were determined by the previous studies (see text)

50 K Naito, T Maruyama with the upper trunk reference frame in three directions In this case, the unit vectors corresponding to the directions of Z, y 0 and x 00 were defined as the joint axes of the shoulder horizontal adduction/abduction (s 1 ), adduction/abduction (s 2 ) and external/internal rotation (s 3 ), respectively For the elbow, the extension (+) and flexion (-) angle h 4 was defined as the Eulerian angle q 2 about y 0 The neutral position of the forearm reference frame was coincident with the upper arm reference frame in three directions In this case, the unit vectors corresponding to the directions of y 0 were defined as the joint axis of elbow extension/flexion (s 4 ) For the wrist, the angles of the ulnar (+) and radial (-) flexion h 5, flexion (+) and extension (-) h, and supination (+) and pronation (-) h 7 were defined as the Eulerian angles q 1, q 2 and q 3 about Z, y 0 and x 00, respectively The neutral position of the hand reference frame was coincident with the forearm reference frame in three directions In this case, the unit vectors corresponding to the directions of Z, y 0 and x 00 were defined as the joint axes of the ulnar/radial flexion (s 5 ), flexion/extension (s ), and supination/pronation (s 7 ), respectively s R, s P, s Y and s j (j = 1 7) described in Fig 1 are defined as the vectors relative to the global reference frame The throwing arm dynamic model was used to calculate the kinematic and kinetic variables of the upper trunk and throwing arm The angular velocity ðx t Þ and acceleration ð _x t Þ of the upper trunk relative to the global reference frame were obtained as follows: x t ¼ s Y _ hy þ s P _ hp þ s R _ hr ð1þ _x t ¼ s Y hy þ s P hp þ s R hr þ s Y hy _ s P hp _ þðs P hp _ þ s Y hy _ Þ s R hr _ : ð2þ The upper arm angular velocities ðx 1 ; x 2 and x 3 Þ and accelerations ð _x 1 ; _x 2 and _x 3 Þ about s j (j = 1 3), the y', y'' X x' z z'' x'', x Z, z' Y y', y'' Fig 2 Eulerian angle convention The rotational sequence of motion follows the order of coordinate axes Z, y 0 and x 00 Joint angles q 1, q 2 and q 3 were defined as the rotational angles about Z, y 0 and x 00 axes of the right-handed reference frame, respectively x'' x' forearm angular velocity ðx 4 Þ and acceleration ð _x 4 Þ about s 4, and hand angular velocities ðx 5 ; x and x 7 Þ and accelerations ð _x 5 ; _x and _x 7 Þ about s j (j = 5 7) were obtained as follows: x j ¼ x j 1 þ s j _ hj ðj ¼ 1 7Þ ð3þ _x j ¼ _x j 1 þ s j hj þ x j s j _ hj ðj ¼ 1 7Þ: ð4þ Here x 0 (in Eq 3) and _x 0 (in Eq 4) were set to be equal to x t and _x t ; respectively The angular velocities x 3 ; x 4 and x 7 were defined as the angular velocities of the upper arm ðx u Þ; forearm ðx f Þ and hand ðx h Þ relative to the global reference frame, respectively The angular accelerations _x 3 ; _x 4 and _x 7 were defined as the angular accelerations of the upper arm ð _x u Þ; forearm ð _x f Þ and hand ð _x h Þ relative to the global reference frame, respectively Vector pointing from a given joint to another joint was defined as the position vector of the distal point of a most distal segment relative to that joint (which is the proximal point of a most proximal segment) in the linked system The vectors pointing from the shoulder to the wrist ðl u f Þ and distal point of the hand ðl u h Þ; and one from the elbow to the distal point of the hand ðl f h Þ were obtained as follows: L u f ¼ l u þ l f ð5þ L u h ¼ l u þ l f þ l h ðþ L f h ¼ l f þ l h : ð7þ Vector pointing from a given joint to the centre of mass of a segment was defined as the position vector of the point of the centre of mass of a segment relative to that joint The vectors pointing from the shoulder to the centre of mass of the forearm ðl u gf Þand hand ðl u gh Þ; and one from the elbow to the centre of mass of the hand ðl f gh Þ were obtained by replacing l f in Eq 5 with l gf, l h in Eq with l gh and l h in Eq 7 with l gh, respectively The velocity of the shoulder relative to the mid-point of the shoulder line ð_l t Þ and velocities of the distal point relative to the proximal point of the upper arm ð_l u Þ; forearm ð_l f Þ and hand ð_l h Þ are expressed as follows: _l s ¼ x s l s (s = t: upper trunk, u: upper arm, f: forearm and h: hand) ð8þ The velocities of the centre of mass relative to the proximal point of the upper arm ð_l gu Þ; forearm ð_l gf Þ and hand ð_l gh Þ are expressed as follows: _l gs ¼ x s l gs ð9þ (s = u: upper arm, f: forearm and h: hand): The accelerations of the centre of mass of the upper arm (a gu ), forearm (a gf ), hand (a gh ) and the distal point of the

Contributions of the muscular torques during overhand baseball pitching 51 hand (a d ) relative to the global coordinate system were obtained as follows: a gu ¼ a m þ _x t l t þ x t _l t þ _x u l gu þ x u _l gu ð10þ a gf ¼ a m þ _x t l t þ x t _l t þ _x u l u þ x u _l u þ _x f l gf þ x f _l gf a gh ¼ a m þ _x t l t þ x t _l t þ _x u l u þ x u _l u þ _x f l f þ x f _l f þ _x h l gh þ x h _l gh a d ¼ a m þ _x t l t þ x t _l t þ _x u l u þ x u _l u þ _x f l f þ x f _l f þ _x h l h þ x h _l h ð11þ ð12þ ð13þ where a m linear acceleration of the mid-point of the hip line The above kinematic variables were calculated using the joint coordinates from the filming data In this study, lengths of the segments jl s j(s = t, u, f and h) were assumed to be constant and corresponded to mean values of the segment lengths from the analysed data The linear acceleration of the mid-point of the hip line (a m ) was obtained by differentiation using the position vector of the mid-point of the vector pointing from the left to right hip Kinetic variables, which are the forces acting at the centre of mass of the upper arm (f gu ), forearm (f gf ), hand (f gh ) and at the distal end point of the hand exerted by the ball (f d ), are expressed as follows: f gs ¼ m s ða gs gþ (s = t: upper trunk, u: upper arm, f: forearm and h: hand) ð14þ f d ¼ m b ða d gþ ð15þ where g ¼ð0 0 9:8Þ: m t ; m u ; m f ; m h and m b are masses of the upper trunk, upper arm, forearm, hand and ball, respectively The resultant joint torques at the shoulder (N S ), elbow (N E ) and wrist (N W ), and the muscular torques, which are determined as the components about the anatomical joint axis of the resultant joint torque, are expressed as follows: N S ¼ l gu f gu þ I u _x u þ x u I u x u þ L u gf f gf þ I f _x f þ x f I f x f þ L u gh f gh þ I h _x h þ x h I h x h þ L u h f d ð1þ N E ¼ l gf f gf þ I f _x f þ x f I f x f þ L f gh f gh þ I h _x h þ x h I h x h þ L f h f d ð17þ N W ¼ l gh f gh þ I h _x h þ x h I h x h þ l h f d ð18þ T j ¼ s T j N S ðj ¼ 1 3Þ ð19þ T 4 ¼ s T 4 N E ð20þ T j ¼ s T j N W ðj ¼ 5 7Þ ð21þ where j joint number and T j the muscular torque of j-th joint of the throwing arm In order to consider the relationships between torques and kinematics in the system, the equation of the throwing arm dynamic model is described as follow: 2 3 2 3 2 3 2 3 T 1 h 1 _h 1 _h 1 T 2 h 2 _h 2 _h 2 T j ¼ H h j þ R _h j þ C _h j þ G þ F 7 4 5 7 4 5 7 4 5 7 4 5 T 7 h 7 _h 7 _h 7 2 3 2 3 2 3 hr hr _ hr _ 7 7 7 þ P l þ P a 4 h P 5 þ P r 4 _h P 5 þ P c 4 _h P 5 ð22þ h Y _h Y _h Y where H the throwing arm inertia matrix, R the throwing arm joint gyroscopic angular acceleration term, C the throwing arm joint angular velocity term (related to centrifugal and Coriolis forces), G the torque due to gravity, F the torque due to the external force exerted by the ball, P l the torque due to the upper trunk linear acceleration P a, P r and P c are the upper trunk angular acceleration, gyroscopic angular acceleration and angular velocity terms, respectively H, R, C, G and F depend on the throwing arm joint kinematics P l, P a, P r and P c depend on the upper trunk kinematics These matrices are given in detail in the literature [1] The muscular torque T j was defined as the joint torque which was the sum of the moments exerted by the muscles, ligaments, tendons, articular capsules and other connecting tissues acting at the joint The muscular torques were calculated by the inverse dynamics with the inertial properties of the trunk and throwing arm segments presented by Ae et al [2] Equation 22 can now be rewritten as follows: 2 3 2 3 2 3 2 3 h 1 T 1 _h 1 _h 1 h 2 T 2 _h 2 _h 2 h j 7 ¼ H 1 T j 7 H 1 R _h j 7 H 1 C _h j 7 4 h 7 7 5 7 4 5 T 7 4 _h 7 7 5 4 _h 7 7 5 H 1 G H 1 F H 1 P l H 1 7 P a 4 h P 5 h Y 2 3 2 3 hr _ hr _ H 1 7 P r 4 _h P 5 H 1 7 P c 4 _h P 5: ð23þ _h Y _h Y Equation 23 expresses the cause effect relationships between the throwing arm joint angular accelerations and the terms due to the muscular torque, throwing arm joint motion, gravity, external force and upper trunk motion 2 hr 3

52 K Naito, T Maruyama Here, the fourth joint angular acceleration h 4 indicates elbow extension/flexion angular acceleration As elbow joint motion was the focus of this study, only the relationships between the elbow joint extension/flexion motion and each dependent component were analysed as the following two steps First, each dependent component of the elbow extension/flexion angular acceleration was obtained using Eq 23 In order to obtain each dependent component, the kinematic and kinetic variables from the experimental trials were input to the right side of Eq 23 Second, each dependent component of the elbow extension/flexion angular acceleration was integrated forward in time during the elbow extension, defined as the period between the instant of the initiation of the elbow extension and the instant of the maximum elbow extension angular velocity By means of these two steps, the throwing arm muscular-, joint motion (gyroscopic angular acceleration and angular velocity)-, gravity-, external force and upper trunk motion (linear acceleration, angular acceleration, gyroscopic angular acceleration and angular velocity)-dependent components developing the maximum elbow extension angular velocity during the elbow extension were obtained 3 Results For all pitchers, the mean values of kinematic, temporal variables and contributions were calculated The mean ball speed was 341±08 m/s, as measured by the radar gun All the normalised elbow extension angular velocities, where 0% corresponded to the instant of the stride foot contact (SFC) and 100% corresponded to the instant of the ball release (REL), are shown in Fig 3 The temporal parameters of the instant of SFC, REL and the occurrence of the maximum elbow extension angular velocity were defined as in the previous study [10], and the occurrence of maximum elbow extension angular velocity was 94 ± 1% In all trials, the maximum elbow extension angular velocity occurred just before ball release The mean maximum elbow extension angular velocity was 379 ± 32 rad/s The typical patterns of the angular acceleration, muscular torque- and motion-dependent angular accelerations of the elbow extension/flexion are presented in Fig 4 EE, MER and MEV in the figures represent the events defined as the instant of the beginning of the elbow extension, maximum shoulder external rotation and maximum elbow extension angular velocity, respectively Here the instant of the beginning of the elbow extension (EE) was determined as the instant of the maximum elbow flexion before the elbow extension motion According to the other studies [3, 10, 18], the acceleration phase during pitching is defined as the period from MER to REL Angular velocity(rad/s) 40 30 20 10 0-10 -20 SFC REL 0 20 40 0 80 100 % Pitch Fig 3 Normalized elbow extension/flexion angular velocities of all trials 0% corresponds to the instant of the stride foot contact (SFC) and 100% corresponds to the instant of the ball release (REL) The elbow extension angular acceleration increasingly occurred from before EE until MEV (Fig 4a) and it primarily resulted from the throwing arm joint and upper trunk angular velocity-dependent components (Fig 4b) Both kept acting generated extension angular acceleration until REL The muscular torque-dependent component largely acted to provide flexion angular acceleration after MER The magnitudes of the upper trunk linear acceleration- and angular acceleration-dependent components were small Also, the throwing arm joint and upper trunk gyroscopic angular accelerations, gravity and external forcedependent components were too small to show them Table 1 shows the contributions of the throwing arm joint muscular torque-, motion-, gravity-, external forceand upper trunk motion-dependent components From these results, it was indicated that the upper trunk angular velocity-dependent component was the greatest contributor, and the throwing arm joint angular velocity-dependent component was the second largest contributor None of the other factors exceeded a 10% contribution The throwing arm joint muscular torque-dependent component indicated a large negative value of -449% Table 2 shows the contributions of each joint muscular torque-dependent component of the throwing arm In Table 2, no factor exceeded 30% The elbow extensor/flexor and shoulder horizontal adductor/abductor torque-dependent components generated large negative angular velocities that exceeded -30% Table 3 shows the contribution of each joint angular velocity-dependent component of the throwing arm, and Table 4 shows the one of the trunk In Tables 3 and 4, the upper trunk counterclockwise clockwise rotation angular velocity-dependent component is show to have been the

Contributions of the muscular torques during overhand baseball pitching 53 Table 1 The mean values of the contributions of throwing arm joint muscular-, motion-, gravity-, external force- and the upper trunk motion-dependent components to the maximum elbow angular velocity (n = ) (a) SFC 3000 Extension(+) 2000 Angular acceleration (rad/s 2 ) (b) Angular acceleration (rad/s 2 ) 1000 0-1000 -2000-3000 SFC 3000 Extension(+) 2000 1000 0-1000 -2000-3000 Flexion(-) -015 Flexion(-) -015-010 -010 MER REL -005 000-005 000 Components ; Muscular torque-dependent ( ) Throwing arm joint angular velocity-dependent ( ) Upper trunk linear acceleration-dependent ( ) Upper trunk angular acceleration-dependent ( ) Upper trunk angular velocity-dependent ( ) MEV largest (34%) From these results, it was indicated that the maximum elbow extension angular velocity was mostly generated by the upper trunk counterclockwise/clockwise rotation angular velocity-dependent component In addition, the shoulder horizontal adduction abduction angular velocity-dependent component was the second largest contributor at 33% (Table 3) EE EE MER MEV REL Fig 4 Typical patterns of angular acceleration (a), and muscularand motion-dependent angular accelerations (b) of the elbow extension/flexion Throwing arm joint and upper trunk gyroscopic angular acceleration, gravity and external force-dependent components are not shown in b because of too small magnitudes SFC, EE, MER, MEV and REL denote the events of the throwing movement which are the instant of the stride foot contact, beginning of the elbow extension, maximum shoulder external rotation, maximum elbow extension angular velocity and ball release, respectively Angular velocity (rad/s) SD Contribution (%) Throwing arm joint muscular torque -170 322-449 Throwing arm joint gyroscopic 00 00 00 angular acceleration Throwing arm joint angular velocity 245 5 45 Gravity -21 03-54 External force 04 57 10 Upper trunk linear acceleration 27 35 72 Upper trunk angular acceleration -28 42-74 Upper trunk gyroscopic angular -24 05-4 acceleration Upper trunk angular velocity 347 59 914 Table 2 Mean values of the contribution of the each joint muscular torque-dependent component of the throwing arm to maximum elbow angular velocity (n = ) Angular velocity (rad/s) SD Contribution (%) Shoulder horizontal adductor abductor -128 30-338 Shoulder adductor abductor 28 09 73 Shoulder external internal rotator 75 2 197 Elbow extensor-flexor -142 210-375 Wrist ulnar radial deviator -90 87-238 Wrist flexor extensor 90 127 237 Wrist supinator pronator -02 22-05 Table 3 Mean values of the contribution of the each joint angular velocity-dependent component of the throwing arm to maximum elbow angular velocity (n = ) Angular velocity (rad/s) Shoulder horizontal adduction 128 43 33 abduction Shoulder adduction abduction 09 11 23 Shoulder external internal rotation 10 7 27 Elbow extension flexion 10 33 280 Wrist ulnar radial deviation 01 04 01 Wrist flexion extension -05 07-12 Wrist supination pronation -04 03-11 SD Contribution (%)

54 K Naito, T Maruyama Table 4 Mean values of the contribution of the each joint angular velocity-dependent component of the upper trunk to maximum elbow angular velocity (n = ) 4 Discussion Angular velocity (rad/s) The purpose of this study was to determine the contributions of the muscular torques and rotations of the throwing arm joint and trunk motions in developing maximal elbow extension angular velocity during overarm throwing The research also aimed to answer the question of which joint is primarily responsible for the generation of rapid elbow extension through muscular or motion-dependent torques using experimental results and model analysis A previous study analysed motion-dependent angular acceleration of the throwing forearm [8] Although their model decomposed the joint angular acceleration into the inertial angular acceleration-, velocity-, and linear acceleration-dependent components of the proximal and distal segments, the model was not able to quantify the elbow extension/flexion angular acceleration due to the multijoint actions of the throwing arm and trunk because the two-segment simple model was used For example, the previous model did not present the motion-dependent angular acceleration of the elbow extension/flexion due to the wrist and trunk rotations about three joint axes In contrast, the present model was able to decompose the elbow extension/flexion motion into each joint muscular and motion-dependent component of the throwing arm and upper trunk rotations, including 10 degrees of freedom This study did not directly calculate the upper trunk muscular torque However, the upper trunk angular acceleration-dependent component in this model must be responsible for the function of the upper trunk muscular torque because of the accordance of the direction of the joint axis of which the upper trunk muscular torque and joint angular acceleration resulted from the upper trunk muscular torque The analysis of the present study deals with the whole multi-joint limb system including the throwing arm and trunk As the present model enables us to determine the causal factors for producing the rapid elbow extension in detail, it is more informative and useful to identify the determinant causes of rapid elbow extension than the two kinetic chain analysis In the previous studies, the question of how the rapid elbow extension during overarm throws is generated via SD Posterior/anterior lean 57 23 150 Medial/lateral lean 49 22 129 Counterclock wise/clockwise rotation 240 47 34 Contribution (%) muscular- and/or motion-dependent interactions has been discussed Feltner [] computed the motion-dependent components due to the upper arm inertial angular velocity, acceleration and the shoulder linear acceleration using two-segment kinetic chain model In that study, it was shown that the magnitude of the upper arm angular velocity-dependent component of the elbow extension angular acceleration remained large (exceeded 15,000 per s 2 ) from the time that the elbow joint began to extend (t = 985 s) until the time the elbow extension angular velocity increased rapidly (t = 995 s) as shown in Fig b of the literature Although the contributions of each factor were not shown in that study, it was strongly suggested in the graph that the elbow extension angular acceleration resulted mostly from the upper arm angular velocity-dependent component Feltner concluded that inertial angular velocity of the upper arm, due primarily to the rotation of the trunk, was the main contributor to the elbow extension angular acceleration during the early phase of the pitching motion Additionally, both shoulder horizontal abduction angular acceleration and the backward acceleration of the throwing shoulder were indicated to contribute to elbow extension angular acceleration near the initiation of elbow extension A computer simulation study of overarm throws in the sagittal plane proved that the fastest throws were achieved by torque reversal at the proximal joints occurring late in the throw and temporal onset of the muscular torque in a proximal to distal sequence [11] This study also showed that the greater magnitude of torque reversal at the proximal joint enhances the angular velocity of the distal segment The effects of the shoulder horizontal abduction angular acceleration, backward acceleration at the throwing shoulder defined by Feltner [] and the simulation study seem to suggest that reversing muscular torque by proximal muscles was advantageous to increase the rapid elbow extension angular velocity and enhance ball speed On the other hand, a separate study [14] indicated that a reversing muscular torque which would enhance the ball velocity and a P D sequence of the temporal onset of the proximal muscular torques, including the trunk rotator and shoulder horizontal adductor, were not observed Hong et al [14] did not agree with the notion that rapid elbow extension in baseball pitching is caused by torque reversal of proximal trunk and shoulder muscles From the results of the contributions in this study, it was shown that the upper trunk counterclockwise/clockwise angular velocity-dependent component was the main contributor (greater than 0%) In addition, the shoulder horizontal adduction/abduction angular velocity-dependent component was the second largest contributor (33%) As the upper trunk rotates in the counterclockwise direction and the shoulder joint adducts close to horizontally during

Contributions of the muscular torques during overhand baseball pitching 55 acceleration phase in pitching [5], both contributions must be due to the angular velocities of the upper trunk counterclockwise rotation and shoulder horizontal adduction On the other hand, the contribution of the throwing arm joint muscular torque-dependent component was negative (-449%, Table 1) Especially, the elbow extensor/flexor- and shoulder horizontal adductor/abductor-dependent components were shown as large negative values that exceeded -30% (Table 2) As the elbow extensor/flexor torque-dependent component could be directly responsible for the torque due to the elbow extensor/flexor muscles resulting in the elbow extension/ flexion joint angular acceleration, the large negative effect of the elbow extensor/flexor-dependent component could represent the action of the elbow flexor torque act to flex during acceleration phase (Fig 4) The negative effect of the elbow flexor is acceptable because the other studies reported that the elbow flexor torque occurred before ball release [, 7, 9, 18] In addition to the large negative value of shoulder horizontal adductor/abductor-dependent component, the contribution of the upper trunk angular acceleration, responsible for the function of the upper trunk rotator, showed a negative value (-74%, Table 1) Therefore, it is concluded that the rapid elbow extension was primarily produced by non-muscular torques due to the upper trunk counterclockwise rotation and the shoulder horizontal adduction angular velocity-dependent component, but not generated by muscular torque reversal at the proximal shoulder or trunk joint These results support the explanation by Feltner [] that the rapid elbow extension during baseball pitching was due primarily to the counterclockwise rotation of the trunk and not from the action of the elbow extensor muscles The large contribution of the upper trunk counterclockwise rotation and shoulder horizontal adduction angular velocity-dependent components in this study suggest an increase in the centrifugal forces acting at the shoulder and elbow joint due to the rotation of the upper trunk and upper arm Stodden et al [18] showed that increased proximal force at the shoulder and elbow joints is significantly associated with increased ball velocity They discussed that the increased proximal force at the shoulder and elbow joints may be highly correlated with the increased centrifugal forces acting at the shoulder and elbow joints The relationship between the increased proximal forces at the shoulder and elbow joint and the increased ball velocity support the findings of the present study The finding that the rapid elbow extension during baseball pitching is primarily due to the angular velocitydependent torques of the upper trunk counterclockwise rotation and shoulder horizontal adduction leads to the suggestion that maintaining the angular velocity of the upper trunk counterclockwise rotation and shoulder horizontal adduction could be required to generate rapid elbow extension Hong et al [14] reported that the torques resulting from the shoulder and elbow joint forces have a tendency to produce the negative torque that decelerate the angular velocities of the upper trunk counterclockwise rotation and shoulder horizontal adduction during the period from before EE to shortly before REL, while the upper trunk counterclockwise rotator and shoulder horizontal adductor counteract negative torques to increase the angular velocities of the counterclockwise rotation and horizontal adduction Because the maximum elbow extension angular velocity occurs just before REL [18], the trunk counterclockwise rotator and shoulder horizontal adductor are partly required to generate positive torques in order to prevent the slowing of the trunk counterclockwise rotation and shoulder horizontal adduction This study found that the angular velocity-dependent interactions due to the upper trunk counterclockwise rotation and horizontal adduction contribute to the production of the rapid elbow extension This finding agrees with the fact that major muscles in natural forceful throws, involving upper trunk counterclockwise rotator and shoulder horizontal adductor, keep acting until shortly before ball release without reversal of proximal shoulder and trunk muscles [14] This study implies that positive torques to maintain the angular velocities of the upper trunk counterclockwise rotation and shoulder horizontal adduction, due to the upper trunk counterclockwise rotator and shoulder horizontal adductor, may play a key role in producing the rapid elbow extension Here it should be noted that the main focus in this study is to improve the model and answer how the rapid elbow extension motion during baseball pitching is produced by dynamic sense using the experiment data and the improved model As discussed above, the present model is helpful to explain the relationships between the experimental results and dynamic factors However, the small sample size of this study just three pitchers makes applying the conclusions of this study to the wider population of various pitchers difficult Although the contributors in this study strongly substantiate the concept that the rapid elbow extension that occurred before ball release could result from the proximal joint motiondependent interaction and not from the actions of the elbow extensor muscles [], it is necessary to repeat the study on a large sample of pitchers to determine the exact qualifications of each contributor In the future, to compare the contributions between skillful and unskillful throwers, it will be interesting to examine more samples of various level throwers

5 K Naito, T Maruyama 5 Conclusion This study showed that rapid elbow extension was primarily due to the upper trunk counterclockwise rotation and shoulder horizontal adduction angular velocitydependent torques, not due to the muscular torque reversal at the proximal shoulder or trunk joint Thus, the trunk counterclockwise rotator and shoulder horizontal adductor may play a key role in producing the rapid elbow extension The present model is helpful to determine the contributions of causal factors in the rapid elbow extension during baseball pitching and understand the underlying mechanism of the throwing motion References 1 Abdel-Aziz YI, Karara HM (1971) Direct linear transformation from comparator coordinates into object-space coordinates in close-range photogrammetry In: Proceedings of the ASP UI symposium on close range photogrammetry American Society of Photogrammetry, Falls Church, VA, pp 1 19 2 Ae M, Tang HP, Yokoi T (1992) Estimation of inertia properties of the body segments in Japanese athletes Biomechanisms 11:23 33 (in Japanese) 3 Barrentine SW, Matsuo T, Escamilla RF, Fleisig GS, Andrews JR (1998) Kinematic analysis of wrist and forearm during baseball pitching J Appl Biomech 14:24 39 4 Chao EY (1980) Justification of triaxial goniometer for the measurement of joint rotation J Biomech 13:989 100 5 Escamilla RF, Fleisig GS, Barrentine SW, Zheng N, Andrews JR (1998) Kinematic comparisons of throwing different types of baseball pitches J Appl Biomech 14:1 23 Feltner ME (1989) Three-dimensional interactions in a two-segment kinetic chain Part II: Application to the throwing arm in baseball pitching Int J Sport Biomech 5:420 450 7 Feltner ME, Dapena J (198) Dynamics of the shoulder and elbow joints of the throwing arm during a baseball pitch Int J Sport Biomech 2:235 259 8 Feltner ME, Dapena J (1989) Three-dimensional interactions in a two-segment kinetic chain Part I: General model Int J Sport Biomech 5:403 419 9 Fleisig GS, Steve WB, Escamilla RF (199) Biomechanics of the elbow in the throwing athelete Oper Tech Sports Med 4:2 8 10 Fleisig GS, Steve WB, Zheng N (1999) Kinematic and kinetic comparison of baseball pitching among various levels of development J Biomech 32:1371 1375 11 Herring RM, Chapman AE (1992) Effects of changes in sequential values and timing of both torque and torque reversal in simulated throws J Biomech 25:1173 1184 12 Hirashima M, Kudo K, Ohtsuki T (2003a) Utilization and compensation of interaction torques during ball-throwing movements J Neurophysiol 89:1784 179 13 Hirashima M, Ohgane K, Kudo K, Ohtsuki T (2003b) Counteractive relationship between the interaction torque and muscle torque at the wrist is predestined in ball-throwing J Neurophysiol 90:1449 143 14 Hong DA, Cheung TK, Roberts EM (2001) A three-dimensional, six-chain analysis of forceful overarm throwing J Electromyogr Kinesiol 11:95 112 15 Joris HJJ, Edwards van Muyen AJ, van Ingen Schenau GJ, Kemper HCG (1985) Force, velocity and energy flow during the overarm throw in female handball players J Biomech 18:409 414 1 Naito K (2007) Analysis of the cause effect relationship of human motion using the biomechanical models of running, hopping and throwing Doctoral dissertation, Tokyo Institute of Technology (in Japanese) 17 Putnam CA (1993) Sequential motions of body segments in striking and throwing skills: descriptions and explanations J Biomech 2:125 135 18 Stodden DF, Fleisig GS, MacLean SP, Andrews JR (2005) Relationship of biomechanical factors to baseball pitching velocity: within pitcher variation J Appl Biomech 21:44 5